#268078

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,13,5,4,6,9)(3,8,14,16,11,10,12,15)(17,18,21,26,23,28,22,27)(19,24,30,39,31,40,37,44)(20,32,34,43,36,42,35,25)(29,41,33,47,45,48,46,38);(1,3,5,11)(2,8,4,10)(6,14,7,12)(9,16,13,15)(17,19)(18,24)(20,33)(21,30)(22,37)(23,31)(25,41)(26,39)(27,44)(28,40)(29,35)(32,47)(34,45)(36,46)(38,42)(43,48);(1,4,5,2)(3,10,11,8)(6,9,7,13)(12,15,14,16)(17,20,23,36)(18,25,28,43)(19,29,31,45)(21,35,22,34)(24,38,40,47)(26,42,27,32)(30,46,37,33)(39,48,44,41);(6,7)(9,13)(12,14)(15,16)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175274);